Reachability in temporal graphs under perturbation
Jessica Enright, Laura Larios-Jones, Kitty Meeks, William Pettersson
TL;DR
The paper investigates reachability in temporal graphs under time perturbations, formalizing δ-perturbations and (δ,ζ)-perturbations to model uncertainty in edge times. The authors introduce Temporal Reachability with Limited Perturbation (TRLP) and Temporal Reachability with Perturbation (TRP) to determine whether a perturbation can raise the maximum reachability to at least $h$, and demonstrate a dichotomy: TRLP is generally NP-hard and [2]-hard in $\zeta$, yet becomes tractable when $\zeta \ge h-1$, with efficient knapsack- and treewidth-based dynamic programs for restricted structures. An XP algorithm enumerates perturbation sets in $n^{O(\zeta)}\log \tau(G,\lambda)$ time, and large-perturbation regimes yield polynomial-time solvability. The results highlight a frontier between general hardness and structural tractability, and contrast reachability robustness with related eccentricity problems that remain hard even under perturbations.
Abstract
Reachability and other path-based measures on temporal graphs can be used to understand spread of infection, information, and people in modelled systems. Due to delays and errors in reporting, temporal graphs derived from data are unlikely to perfectly reflect reality, especially with respect to the precise times at which edges appear. To reflect this uncertainty, we consider a model in which some number $ζ$ of edge appearances may have their timestamps perturbed by $\pmδ$ for some $δ$. Within this model, we investigate temporal reachability and consider the problem of determining the maximum number of vertices any vertex can reach under these perturbations. We show that this problem is intractable in general but is efficiently solvable when $ζ$ is sufficiently large. We also give algorithms which solve this problem in several restricted settings. We complement this with some contrasting results concerning the complexity of related temporal eccentricity problems under perturbation.